Terminal Defects, Growing Multiplicity, and Variance Extremality in the Double Dixie Cup Problem

TL;DR

This paper introduces a new terminal-defect method to analyze the double Dixie cup problem, proving variance extremality and deriving new asymptotics for collection times under various probability distributions.

Contribution

It establishes the finite-variance extremality conjecture, proves a growing-multiplicity Gumbel theorem, and extends variance asymptotics to unequal probabilities and growing collection sizes.

Findings

Variance minimized at uniform coupon probabilities

Variance increases along rays from the uniform vector

Asymptotic behavior characterized for unequal probabilities

Abstract

We develop a terminal-defect method for the double Dixie cup problem and use it to prove the finite-variance extremality conjecture of Doumas and Papanicolaou. For every \(m\ge1\) and \(N\ge2\), among all positive coupon probability vectors \(p=(p_1,\ldots,p_N)\), the variance of the time \(T_m(N)\) to collect \(m\) complete sets is uniquely minimized at the uniform vector. We prove the stronger radial statement that the variance is strictly increasing along every ray from the uniform vector. The proof is finite-\(N\) and exact: after Poissonization, the completion time is a maximum of independent Erlang variables, and the radial derivative of its distribution is compared to a size-biased law using a monotone-likelihood-ratio argument based on a log-scale monotonicity property of the Gamma reverse hazard. The same framework gives a growing-multiplicity Gumbel theorem in the equal-probability case, with expectation and variance asymptotics on the inverse gamma-tail scale. This recovers the fixed-\(m\) equal-probability variance asymptotic stated as Conjecture 1 by Doumas and Papanicolaou, classically known for \(m=1\), and extends the mechanism to \(m=m_N\). We also illustrate the unequal-probability theory with endpoint-Laplace limits for power-law probabilities.